A) Easy way to test a starter is to buy a starter switch (see pic below) that attaches to the leads in the back and bypasses all but the power leads. Always good to have one aboard anyway in case your starter switch fails. This will isolate the outside problems and then it's either the motor or solenoid or electric feed. If you remove the starter and hook it up to positive and negative (laying it on a towel or such to keep it from moving if you don't have someone to hold it) and this switch, you can see the Bendix gear snap forward as you push the button. That means the solenoid is OK. The motor will turn with a lot of force (which is why it should be held, etc) after the Bendix gear has thrust forward if you have A) good electricity and/or B) it is OK. If it turns slowly or not at all, you either have a bad connection to it, bad/low batteries or it needs to be rebuilt. You shouldn't need any professional help to go through this process if you are a DIY person. If the motor does not do as above, take it to a shop and they will handle the rebuild.I had to replace my starter that burned up last year (a DIM job). A)
Can't assume that the yard guys have seen and solved every boat problem that comes to them; some are new-like yours. It's better for them to take time and care on diagnostics rather than to f-up the owner's boat or engine. Anything they break they get to fix for "free." This is not necessarily incompetence--it might be "learning curve."
B) I'm sorry, but I completely disagree with this statement. If someone is charging me money and saying they are professionals, they had better NOT be charging me $80 an hour for their education. It is their expertise and knowledge that I am paying THEM for (any $15.00 an hour kid can turn wrenches) and if they aren't able to do the diagnostics, then they either shouldn't charge me for their ineptitude, or bring in a professional who knows his stuff and charge me accordingly.
Way too many "marine professionals" are just parts changers these days, changing parts until they finally get the right one, and it costs the customer a great deal of unnecessary money.
I'm sure Mainesail could weigh in on this one with many many tales of what he's had to do to clean up the messes other "professionals" have made before him, not to mention boat owners out of their depth.